This lab deals with time constants of capacitors in relation to resistance, series versus parallel, and location. The goal was to understand the effects on time constants for different configuration of RC circuits. Capacitors are timing elements in circuits that slowly eliminate the current as they gain charge. Time Constant (τ) is the amount of time it takes for current to reach 37% of the original current. By using the Fourier current sensor, Fourier data collection unit, and a laptop, the time constants can be expressed through graphs in the three sections of this lab. We will first configure the RC circuit using six different resistors (15kΩ, 22kΩ, 27kΩ, 39kΩ, 56kΩ, 68kΩ) and one capacitor (470 µF). Then, we will calculate the time constant with six capacitors in both series and parallel.
Procedure
Part I (Time Constants with Capacitors and Resistors)-
We first mapped out a simple circuit using only one of the six resistors, a 470µF capacitor, an ammeter, and a switch. Then we built the circuit on the breadboard and connected the Fourier current sensor and the Fourier data collection unit. We connected the current sensor in such a way that it was a part of the circuit and set the power supply to 12 volts. After connecting the power supply to the breadboard we began recording data on the computer. Since capacitors eventually eliminate all current in the circuit as it gains charge, the current eventually dwindled down to approximately zero. Once this occurred, we stopped the recording, turned off the switch, and used the data to create a graph. We calculated the actual time constant, the theoretical time constant, and the percent error based on the data. Our group did this entire process six times using one of six different resistors each time: 15,000Ω, 22,000Ω, 27,000Ω, 39,000Ω, 56,000Ω, 68,000Ω.
Part 2(Time Constants with Capacitors in Series vs. Parallel)-
Our next objective was to observe the changes in time constants using capacitors in series and parallel.
Series:
We built the same RC circuit as before with one capacitor and then added a capacitor after each trial until we reached a total of six capacitors in series. Using the Multilab program and the Fourier data collection unit from Part 1, we began collecting data by turning the switch on. Once the current became zero, we stopped the recording and turned off the switch. We compiled all the data into a graph and found the time constant and the effective capacitance each time. After all six capacitors we completed and the time constants for each were accounted for, we moved on to the parallel section.
Parallel:
To analyze time constants with capacitors in parallel, our group performed the same procedure as in series, but the capacitors were arranged in parallel this time. We began with one capacitor and added a capacitor in parallel each time until we reached a total of six capacitors on the breadboard. Once we created a graph using the time constants from our data, we compared this graph to the graph of capacitors in series and observed the differences between the two.
Results
Table 1. Time Constants In Relation to Resistance
Resistor
Actual Time Constant
Theoretical Time Constant
Percent Error
15000
7.6
7.05
7.80%
22000
10.8
10.34
4.45%
27000
13.45
12.69
5.98%
39000
18.6
18.33
1.47%
56000
26.2
26.32
0.45%
68000
33.7
31.96
5.44%
The data suggests that the percent error between the actual and theoretical time constants decreases as the resistance gets higher. The 60 kΩ resistor is most likely a error in the collection method.
Figure 1. Example graph of current versus time
Finding the x-value (time) where the y-value (current) is 37% of the starting value is how we determined the actual time constant.
Figure 2: Time constant for multiple capacitors in series
Figure 3: Time Constant for multiple capacitors in parallel
As can be seen, adding capacitors in series decreased the time constant for the circuit, while adding them in parallel increased the time constant for the circuit.
Conclusions
According to the values in Table 1, a higher resistor value results in a higher the time constant. For example, 15kΩ had an actual time constant of 7.6 seconds whereas the 68kΩ resistor had and actual time constant of 33.7 seconds. Since resistance essentially slows the current, the greater the resistor value, the longer it took the current to flow to the capacitor. The smaller the current flowing to the capacitor, the longer it takes the capacitor to charge because there are less electrons flowing to the capacitor each second. Thus, this resulted in a greater time constant because it took longer for the capacitor to charge up and eliminate the current in the circuit.
When comparing the theoretical time constants to the actual time constants in Table 1, the accuracy was not good. Perhaps our group did not accurately determine the actual time constant. In Figure 1, for example, it is difficult to find the actual time constant on a graph like this because there is gap between sets of data. In addition, another source of error could have been as a result of deviations in resistors. Resistor values are only approximations not exact values, so this could have added potential error in our calculations. In Table 1, although there was one exception, it seemed as if the overall percent error reduced as the resistance increased.
(Part 2)
In series, the time constant increased as more capacitors were added because the current can flow through each one, and the capacitors do not have to be completely filled (they only have to add to the supply voltage). The time constant for one capacitor in series was 0.245 seconds, but the time constant for 6 capacitors in series was 0.03 seconds. However, when more capacitors were added in parallel, the time constant increased because each electron can only flow to one capacitor at a time, and each capacitor in parallel has to be completely filled. Thus, the more capacitors that need to be charged, the longer it takes to fill each one. For instance, the time constant for one capacitor in parallel was 0.25 seconds whereas the time constant for 6 capacitors in parallel was 1.5 seconds. This would suggest that adding resistors in series increases the effective capacitance of the RC circuit while adding capacitors in parallel decreases the effective capacitance. This is a useful piece of information to have if we are trying to design an RC circuit with a specific time constant.
Table of Contents
Capacitor Time Constants
Former Student
Introduction
This lab deals with time constants of capacitors in relation to resistance, series versus parallel, and location. The goal was to understand the effects on time constants for different configuration of RC circuits. Capacitors are timing elements in circuits that slowly eliminate the current as they gain charge. Time Constant (τ) is the amount of time it takes for current to reach 37% of the original current. By using the Fourier current sensor, Fourier data collection unit, and a laptop, the time constants can be expressed through graphs in the three sections of this lab. We will first configure the RC circuit using six different resistors (15kΩ, 22kΩ, 27kΩ, 39kΩ, 56kΩ, 68kΩ) and one capacitor (470 µF). Then, we will calculate the time constant with six capacitors in both series and parallel.
Procedure
Part I (Time Constants with Capacitors and Resistors)-
We first mapped out a simple circuit using only one of the six resistors, a 470µF capacitor, an ammeter, and a switch. Then we built the circuit on the breadboard and connected the Fourier current sensor and the Fourier data collection unit. We connected the current sensor in such a way that it was a part of the circuit and set the power supply to 12 volts. After connecting the power supply to the breadboard we began recording data on the computer. Since capacitors eventually eliminate all current in the circuit as it gains charge, the current eventually dwindled down to approximately zero. Once this occurred, we stopped the recording, turned off the switch, and used the data to create a graph. We calculated the actual time constant, the theoretical time constant, and the percent error based on the data. Our group did this entire process six times using one of six different resistors each time: 15,000Ω, 22,000Ω, 27,000Ω, 39,000Ω, 56,000Ω, 68,000Ω.
Part 2(Time Constants with Capacitors in Series vs. Parallel)-
Our next objective was to observe the changes in time constants using capacitors in series and parallel.
Series:
We built the same RC circuit as before with one capacitor and then added a capacitor after each trial until we reached a total of six capacitors in series. Using the Multilab program and the Fourier data collection unit from Part 1, we began collecting data by turning the switch on. Once the current became zero, we stopped the recording and turned off the switch. We compiled all the data into a graph and found the time constant and the effective capacitance each time. After all six capacitors we completed and the time constants for each were accounted for, we moved on to the parallel section.
Parallel:
To analyze time constants with capacitors in parallel, our group performed the same procedure as in series, but the capacitors were arranged in parallel this time. We began with one capacitor and added a capacitor in parallel each time until we reached a total of six capacitors on the breadboard. Once we created a graph using the time constants from our data, we compared this graph to the graph of capacitors in series and observed the differences between the two.
Results
Table 1. Time Constants In Relation to Resistance
The data suggests that the percent error between the actual and theoretical time constants decreases as the resistance gets higher. The 60 kΩ resistor is most likely a error in the collection method.
Figure 1. Example graph of current versus time
Finding the x-value (time) where the y-value (current) is 37% of the starting value is how we determined the actual time constant.
Figure 2: Time constant for multiple capacitors in series
Figure 3: Time Constant for multiple capacitors in parallel
As can be seen, adding capacitors in series decreased the time constant for the circuit, while adding them in parallel increased the time constant for the circuit.
Conclusions
According to the values in Table 1, a higher resistor value results in a higher the time constant. For example, 15kΩ had an actual time constant of 7.6 seconds whereas the 68kΩ resistor had and actual time constant of 33.7 seconds. Since resistance essentially slows the current, the greater the resistor value, the longer it took the current to flow to the capacitor. The smaller the current flowing to the capacitor, the longer it takes the capacitor to charge because there are less electrons flowing to the capacitor each second. Thus, this resulted in a greater time constant because it took longer for the capacitor to charge up and eliminate the current in the circuit.
When comparing the theoretical time constants to the actual time constants in Table 1, the accuracy was not good. Perhaps our group did not accurately determine the actual time constant. In Figure 1, for example, it is difficult to find the actual time constant on a graph like this because there is gap between sets of data. In addition, another source of error could have been as a result of deviations in resistors. Resistor values are only approximations not exact values, so this could have added potential error in our calculations. In Table 1, although there was one exception, it seemed as if the overall percent error reduced as the resistance increased.
(Part 2)
In series, the time constant increased as more capacitors were added because the current can flow through each one, and the capacitors do not have to be completely filled (they only have to add to the supply voltage). The time constant for one capacitor in series was 0.245 seconds, but the time constant for 6 capacitors in series was 0.03 seconds. However, when more capacitors were added in parallel, the time constant increased because each electron can only flow to one capacitor at a time, and each capacitor in parallel has to be completely filled. Thus, the more capacitors that need to be charged, the longer it takes to fill each one. For instance, the time constant for one capacitor in parallel was 0.25 seconds whereas the time constant for 6 capacitors in parallel was 1.5 seconds. This would suggest that adding resistors in series increases the effective capacitance of the RC circuit while adding capacitors in parallel decreases the effective capacitance. This is a useful piece of information to have if we are trying to design an RC circuit with a specific time constant.
References
There were no references for this report.